Theorem r19.26m 2750

Description: Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by NM, 22-Feb-2004.)

Assertion

Ref	Expression
r19.26m	|- (A.x((x e. A -> ph) /\ (x e. B -> ps)) <-> (A.x e. A ph /\ A.x e. B ps))

Proof of Theorem r19.26m

Step	Hyp	Ref	Expression
1		19.26 1593	. 2 |- (A.x((x e. A -> ph) /\ (x e. B -> ps)) <-> (A.x(x e. A -> ph) /\ A.x(x e. B -> ps)))
2		df-ral 2620	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
3		df-ral 2620	. . 3 |- (A.x e. B ps <-> A.x(x e. B -> ps))
4	2, 3	anbi12i 678	. 2 |- ((A.x e. A ph /\ A.x e. B ps) <-> (A.x(x e. A -> ph) /\ A.x(x e. B -> ps)))
5	1, 4	bitr4i 243	1 |- (A.x((x e. A -> ph) /\ (x e. B -> ps)) <-> (A.x e. A ph /\ A.x e. B ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   e. wcel 1710  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ral 2620

This theorem is referenced by: (None)